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Bruce Blackadar

Publications and source records attributed to Bruce Blackadar.

11 recordsLinked to original sources
arXiv2026

Separable C*-algebras Without the Countable Axiom of Choice

The goal of this paper is twofold. In addition to the results stated in the next paragraph, we present some classical results on absoluteness relevant to functional analysis that are well known to logicians but not nearly as well advertised as they should be. We show that the theory of separable C*-algebras can be developed in ZF (that is, without using any Choice). This includes proving the Gelfand-Naimark representation theorems as well as the Spectral Mapping Theorem for polynomials and developing continuous functional calculus for commuting normal elements. Some of our proofs are modifications of the standard ones, obtained by avoiding the use of Choice. Some other proofs require new ideas in order to avoid the use of Choice. Yet another batch of proofs proceeds by using the set-theoretic Shoenfield Absoluteness Theorem. This result (well known to logicians but regrettably not as well advertised as it deserves) implies that statements about standard Borel spaces of low quantifier complexity that are provable in ZFC, or even ZFC together with the Continuum Hypothesis are provable in ZF. One of the main objectives of this paper is to present these results in a convenient form that can be utilized by analysts not familiar with set theory. We also show that in the absence of Choice (more precisely, assuming the existence of a Russell set) there is a concretely representable unital commutative \cstar-algebra that is not isomorphic to C(X) for any compact Hausdorff space X. Finally, from the model-theoretic point of view, while the property of having a tracial state is provably axiomatizable in ZFC, it is not provably axiomatizable in ZF+DC.

math.OA↗
arXiv2023

Hilbert Spaces Without Countable AC

This article examines Hilbert spaces constructed from sets whose existence is incompatible with the Countable Axiom of Choice (CC). Our point of view is twofold: (1) We examine what can and cannot be said about Hilbert spaces and operators on them in ZF set theory without any assumptions of Choice axioms, even the CC. (2) We view Hilbert spaces as ``quantized'' sets and obtain some set-theoretic results from associated Hilbert spaces.

math.LO↗
arXiv2018

A bump in the road in elementary topology

We observe a subtle and apparently generally unnoticed difficulty with the definition of the relative topology on a subset of a topological space, and with the weak topology defined by a function.

math.GN↗
arXiv2015

Lifting Commutation Relations in Cuntz Algebras

We examine splitting of the quotient map from the full free product $A*B$, or the unital free product $A*_{\mathbb C}B$, to the (maximal) tensor product $A\otimes B$, for unital C*-algebras $A$ and $B$. Such a splitting is very rare, but we show there is one if $A$ and $B$ are both the Cuntz algebra $O_2$ or $O_\infty$, and in a few other cases. The splitting is not explicit (and in principle probably cannot be). We also describe severe $K$-theoretic obstructions to a splitting.

math.OA↗
arXiv2015

A General Implicit/Inverse Function Theorem

The Implicit and Inverse Function Theorems are special cases of a general Implicit/Inverse Function Theorem which can be easily derived from either theorem. The theorems can thus be easily deduced from each other via the generalized version.

math.CA↗
arXiv2015

The homotopy lifting theorem for semiprojective C*-algebras

We prove a complete analog of the Borsuk Homotopy Extension Theorem for arbitrary semiprojective C*-algebras. We also obtain some other results about semiprojective C*-algebras: a partial lifting theorem with specified quotient, a lifting result for homomorphisms close to a liftable homomorphism, and that sufficiently close homomorphisms from a semiprojective C*-algebra are homotopic.

math.OA↗
arXiv2012

Extending continuous functions

We examine conditions on a (compact metrizable) space $X$ such that for any space $Y$ and closed subspace $Z$, the set of continuous functions from $Z$ to $X$ which extend to $Y$ is either open or closed in the set of continuous functions from $Z$ to $X$ in the uniform topology.

math.GN↗
arXiv2010

An algebraic approach to the radius of comparison

The radius of comparison is an invariant for unital C*-algebras which extends the theory of covering dimension to noncommutative spaces. We extend its definition to general C*-algebras, and give an algebraic (as opposed to functional-theoretic) reformulation. This yields new permanence properties for the radius of comparison which strengthen its analogy with covering dimension for commutative spaces. We then give several applications of these results. New examples of C*-algebras with finite radius of comparison are given, and the question of when the Cuntz classes of finitely generated Hilbert modules form a hereditary subset of the Cuntz semigroup is addressed. Most interestingly, perhaps, we treat the question of when a full hereditary subalgebra B of a stable C*-algebra A is itself stable, giving a characterization in terms of the radius of comparison. We also use the radius of comparison to quantify the least n for which a C*-algebra D without bounded 2-quasitraces or unital quotients has the property that M_n(D) is stable.

math.OA↗
arXiv2007

Irreducible representations of inner quasidiagonal C*-algebras

It is shown that a separable C*-algebra is inner quasidiagonal if and only if it has a separating family of quasidiagonal irreducible representations. As a consequence, a separable C*-algebra is a strong NF algebra if and only if it is nuclear and has a separating family of quasidiagonal irreducible representations. We also obtain some permanence properties of the class of inner quasidiagonal C*-algebras.

math.OA↗